Optimal filters

At the first eonferenee of the International Federation of Automatie Control (IFAC), Kalman (1960) introdueed the dual eoneept of eontrollability and observability. At the same time Kalman demonstrated that when the system dynamie equations are linear and the performanee eriterion is quadratie (LQ eontrol), then the mathematieal problem has an explieit solution whieh provides an optimal eontrol law. Also Kalman and Buey (1961) developed the idea of an optimal filter (Kalman filter) whieh, when eombined with an optimal eontroller, produeed linear-quadratie-Gaussian (LQG) eontrol. 

Anderson, B.D.O. and Moore, J.B. (1979) Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ. 

Cook, L. N. Laboratory Approach Optimizes Filter-Aid Addition. Chem Eng, July 23, 1984 45-50. 

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

Konstandopoulos, A. G., Skaperdas, E., Warren, J., and Allanson, R. Optimized filter design and selection criteria for continuously regenerating diesel particulate traps. SAE Technical Paper No. 1999-01-0468 (1999). 

The optimal filtering problem (the Kalman-Bucy filter) can be solved independent of the optimal control for the LQP and provides a means for estimating unmeasured state variables which may be corrupted by process and instrument noise. 

After mounting, the slides should be left in the dark for at least 15 minutes for DAPI to stain the nuclear material. Slides are inspected in a fluorescence microscope. It is important to use the recommended filter sets since use of sub-optimal filters significantly reduces signal intensity (see Chapter 12, Filters for FISH Imaging). 

The most common applications of methods for handling sequential series in chemistry arise in chromatography and spectroscopy and will be emphasized in this chapter. An important aim is to smooth a chromatogram. A number of methods have been developed here such as the Savitsky-Golay filter (Section 3.3.1.2). A problem is that if a chromatogram is smoothed too much the peaks become blurred and lose resolution, negating the benefits, so optimal filters have been developed that remove noise without broadening peaks excessively. 

The filter is moved along the time series or spectrum, each datapoint being replaced successively by the corresponding filtered datapoint. The optimal filter depends on the noise distribution and signal width. It is best to experiment with a number of different filter widths. 

The best resolution which can be achieved with optimal filtering does not depend on the value of the resistive chain R, but solely on the eletrode capacitance. The optimum resolution is given by... 

The prior estimates of X and P allow the n 1 Kalman gain K to be calculated through Eqn.(5), where R(k) is the covariance of the kth measurem it. It is worth nothing that the elements of K(k) are computed as the optimal filter weights producing the minimum variance fit of the model to the data. Moreover, the invoted quantity in Eqn.(5) is scalar, so only the reciprocal must be calculated. 

Optimal filtering was proposed by Altman and Jardetzky (1989) as a heuristic refinement method of structure determination and has also been applied to the dihedral angle space (KoeU et al., 1992). Optimal filtering uses the exclusion paradigm, and during the search aU possible conformations are retained except where they are incompatible with the data. This allows a more systematic search of the allowed conformational space. As in the case of distance geometry, it is a ptire geometric method, and it calculates the mean positions and standard deviations of each atom. The output also needs to be refined to add information fi om the empirical force field. 

NMR solution structures, when compared with crystal structures, are less well defined. This is because NMR experiments are done in solution and at room temperatures. Brownian motion of proteins is observed. When a family of structures is calculated, we use the spread of different conformations within the family to represent the precision of the coordinates of the atoms. When optimal filtering (the Kalman filter) is used, the output automatically gives a measure of uncertainty by giving the standard deviation as well as the mean value of the coordinates. 

The optimal number of retained wavelet coefficients, and the reconstruction error strongly depend on the applied filter. It means that the MDL cost function can be used for the filter optimization. Filter for which the MDL achieves minimal value or the minimal number of retained wavelet coefficients can be considered the optimal one for data compression. 

In the discussed approach to data set compression there is a need to predefine percentage of variance to be preserved by the retained coefficients (or alternatively the average RMS of data reconstruction). We performed all calculations for 99% of variance. The best-basis was calculated for all studied filters using entropy as basis selection criterion. As expected, the number N of retained wavelet coefficients, depends on the applied filter. The number of the retained coefficients varies from 139 to 177. The worst compression is with filter No. 2, and the best with filter No. 15 (see Fig. 10). Final results for the optimal filter (No. 15) are presented in Table 2. 

At this point we should note that neither the discrete nor the conventional version of the SPMC filter was used in the work that we review in Section IV. In fact, for the problems discussed below, it is possible to sum up a much larger subset of configurations analytically by exploiting certain symmetry properties. The resulting blocking scheme based on this exact analytical summation is what we referred to earlier as the optimized filter. This technique is discussed in great detail in the following section and in Refs. 27 and 29. 

Optimal filter cake thickness in a batch filter, for which the rate of filtration is balanced against the cost of removing the cake. 

While (Baykut et cd. 2000) have used Random Markov Fields models for defect inspection of textile surfaces. (Kmnar Pang 2002h) used linear FIR filters with optimized energy separation for defect detection and evaluated the performance of different feature separation criteria. They also discussed the design of optimal filters for supervised and imsupervised inspection systems. 

Kumar, A. Pang, G. K. H. 2002b. Defect detection in textured materials using optimized filters. IEEE... 

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